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Chapter 10: Problem 34

Solve the given equation. $$8=3-\sqrt{u}$$

Short Answer

Expert verified
u = 25

Step by step solution

01

Isolate the square root term

Move the constant term on the right-hand side to the left-hand side of the equation in order to isolate the square root term. Subtract 3 from both sides of the equation: \[8 - 3 = 3 - \sqrt{u} - 3\] which simplifies to: \[5 = -\sqrt{u}\]
02

Multiply by -1

Multiply both sides of the equation by -1 to get rid of the negative sign in front of the square root: \[5 \cdot (-1) = -\sqrt{u} \cdot (-1)\] which simplifies to: \[-5 = \sqrt{u}\]
03

Square both sides

Square both sides of the equation to eliminate the square root: \[(-5)^{2} = (\sqrt{u})^{2}\] which simplifies to: \[25 = u\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the variable
To solve an equation with a square root, the first step is often to isolate the term that contains the square root. This means we want to get the square root expression by itself on one side of the equation.

In our example, we start with the equation \(8 = 3 - \sqrt{u}\). The square root term is \( \sqrt{u} \), and it's on the right side of the equation along with a constant 3. To isolate \( \sqrt{u} \), we need to move 3 to the other side.

We do this by subtracting 3 from both sides of the equation. This gives us:
\[8 - 3 = 3 - \sqrt{u} - 3\]
This simplifies to:
\[5 = -\sqrt{u}\]
Now, we have successfully isolated the square root term.
Multiplying by -1
In some equations, after isolating the square root term, you might end up with a negative sign in front of it. To make it easier to work with, we often multiply both sides of the equation by -1.

In our case, we have the equation \(5 = -\sqrt{u}\). To get rid of the negative sign in front of \( \sqrt{u} \), we multiply both sides by -1:
\[5 \cdot (-1) = -\sqrt{u} \cdot (-1)\]
This simplifies to:
\[-5 = \sqrt{u}\]
Now the square root term is positive, and we are one step closer to solving for the variable \( u \).
Squaring both sides
The final step to solve an equation involving a square root is to square both sides. This will eliminate the square root and allow us to solve for the variable directly.

In our current equation, we have \(-5 = \sqrt{u}\). To remove the square root, we square both sides of the equation:
\[(-5)^{2} = (\sqrt{u})^{2}\]
This simplifies to:
\[25 = u\]
Thus, we find that \(u = 25\). We've solved the equation by isolating the square root term, simplifying the equation by multiplying by -1, and finally squaring both sides to find the value of the variable.

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Most popular questions from this chapter

Rationalize the denominators and simplify. $$(4-\sqrt{7})^{2}-\frac{18}{4+\sqrt{7}}$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$3(2 \sqrt{5}+4 \sqrt{10})-5(\sqrt{5}-3 \sqrt{10})$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\sqrt{50 t^{7}}-\sqrt{32 t^{7}}$$

Use a calculator to find the following square roots. Round off your answers to the nearest hundredth. $$\sqrt{2.09}$$

Reduce to lowest terms. $$\frac{4+\sqrt{8}}{6}$$
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